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1 Data

library(fitplc2)

set.seed(123)

SX0 <- 30
PX0 <- 2
X <- 50
xx <- seq(0.01, 5, length = 100)
yy <- fweibull(xx, SX0, PX0, X=50)
yy2 <- rnorm(100, yy, 0.01)

dat <- tibble(relK = yy2, WP = xx) %>%
  filter(relK > 0 & relK < 1) %>%
  mutate(PLC = 100 - 100 * relK)

dat
## # A tibble: 99 x 3
##     relK     WP   PLC
##    <dbl>  <dbl> <dbl>
##  1 0.994 0.01   0.568
##  2 0.996 0.0604 0.392
##  3 0.992 0.161  0.812
##  4 0.987 0.212  1.28 
##  5 0.997 0.262  0.318
##  6 0.977 0.312  2.29 
##  7 0.952 0.363  4.81 
##  8 0.949 0.413  5.11 
##  9 0.942 0.464  5.81 
## 10 0.948 0.514  5.16 
## # … with 89 more rows

2 PLC

\[ PLC = \frac{K_{max} - K}{K_{max}} \cdot 100 \]

where \(K\) is raw conductance, \(K_{max}\) is the maximum value of \(K\).

3 Functions

3.1 Weibull

\[ \frac{K}{K_{max}} = \Biggl(1 - \frac{X}{100}\Biggr)^{\Biggl[{\biggl(\frac{P}{P_X}}\biggr)^{\frac{P_X S_X}{V}}\Biggr]} \]

\[ V = (X-100)\text{ln}\Biggl(1-\frac{X}{100}\Biggr) \]

where \(P_X\) is the xylem pressure (P) where X% of the conductivity is lost, \(S_X\) is the derivative (% MPa-1) at X (e.g. \(S_{50}\) is the slope of the curve at \(P_{50}\)).

fit1 <- fitplc2::fitplc(dat,
                        varnames = c(PLC = "PLC", WP = "WP"),
                        nboot=100)

fit1
## Class of object 'plcfit' as returned by 'fitplc'.
## -------------------------------------------------
## 
## Parameters and %s%% confidence interval:
## 
##  95%    Estimate Norm - 2.5% Norm - 97.5% Boot - 2.5% Boot - 97.5%
## SX 29.858981   29.586862     30.13437   29.570310    30.193488
## PX  2.000767    1.991328      2.01020    1.990307     2.009014

3.2 Invere-Weibull

\[ P = \Biggl[\frac{\text{ln}K}{\text{ln}\bigl(1-\frac{X}{100}\bigr)}\Biggr]^{\frac{V}{P_X S_X}} P_X \]

fit2 <- fitplc2::fitplc(dat,
                        model = "Inv_Weibull",
                        varnames = c(PLC = "PLC", WP = "WP"),
                        nboot=100)
fit2
## Class of object 'plcfit' as returned by 'fitplc'.
## -------------------------------------------------
## 
## Parameters and %s%% confidence interval:
## 
##  95%    Estimate Norm - 2.5% Norm - 97.5% Boot - 2.5% Boot - 97.5%
## SX 29.845611   29.569021     30.13129    29.36082    30.335142
## PX  2.004632    1.980742      2.02845     1.98900     2.015076

3.3 Sigmoidal

\[ K = \frac{1}{1 + exp(a(P - b))} \]

where \(a\) and \(b\) are constants.

fit3 <- fitplc2::fitplc(dat,
                        model = "nls_sigmoidal",
                        varnames = c(PLC = "PLC", WP = "WP"),
                        nboot=100)
fit3
## Class of object 'plcfit' as returned by 'fitplc'.
## -------------------------------------------------
## 
## Parameters and %s%% confidence interval:
## 
##  95%    Estimate Norm - 2.5% Norm - 97.5% Boot - 2.5% Boot - 97.5%
## SX 32.171341   31.060857    33.281825   31.104104    33.259805
## PX  2.093105    2.064374     2.121836    2.072157     2.124412

3.4 Inverse-sigmoidal (logit)

\[ P = \frac{log(\frac{1}{K} - 1)}{a} + b \]

fit4 <- fitplc2::fitplc(dat,
                        model = "Inv_nls_sigmoidal",
                        varnames = c(PLC = "PLC", WP = "WP"),
                        nboot=100)
fit4
## Class of object 'plcfit' as returned by 'fitplc'.
## -------------------------------------------------
## 
## Parameters and %s%% confidence interval:
## 
##  95%    Estimate Norm - 2.5% Norm - 97.5% Boot - 2.5% Boot - 97.5%
## SX 38.134968   35.921749    40.348187   33.675849    42.539186
## PX  2.349362    2.268337     2.430388    2.259403     2.449051

4 Reference

  • Duursma R.A., Choat B. 2017. fitplc - an R package to fit hydraulic vulnerability curves. Journal of Plant Hydraulics. doi:10.20870/jph.2017.e002

  • Ogle, K. , Barber, J. J., Willson, C. and Thompson, B. 2009, Hierarchical statistical modeling of xylem vulnerability to cavitation. New Phytologist, 182: 541-554. doi:10.1111/j.1469-8137.2008.02760.x

  • Pammenter NW, Van der Willigen CV. 1998. A mathematical and statistical analysis of the curves illustrating vulnerability of xylem to cavitation. Tree Physiology 18: 589–593. https://doi.org/10.1093/treephys/18.8-9.589